inequality involving the fractional Sobolev space

Question

Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality $$|u(x)| \leq C \|u\|_{X_{0}} $$ where $u\in X_{0}$, $H^s(\mathbb R)= W^{s, 2}(\mathbb R)$ and $$ \|u\|_{X_{0}}^2= \int_{\mathbb R} |(-\Delta)^{s/2} u|^2dx.$$ Does a weaker version of the inequality exists when $s=1/2.$

Answer 1

In the first place, you must have $s>1/2$. Next you write $$ u(x)=\int_{\mathbb R} e^{2πix\xi}\underbrace{\hat u(\xi)(1+\xi^2)^{s/2}}_{\in L^2(\mathbb R)}\underbrace{(1+\xi^2)^{-s/2}}_{\in L^2(\mathbb R)}d\xi, $$ entailing $ \vert u(x)\vert\le \Vert u\Vert_{H^s(\mathbb R)}\left(\int_{\mathbb R}(1+\xi^2)^{-s} d\xi\right)^{1/2}. $ Moreover, you have with $D_x=-i\partial_x$ $$ 2\Re\langle D_x u, i x u\rangle_{L^2(\mathbb R)}=\Vert u\Vert^2_{L^2(\mathbb R)}, $$ so that if $u$ is supported in $(-1/2, 1/2)$, we have $ \Vert u\Vert^2_{L^2(\mathbb R)}\le \Vert u\Vert_{L^2(\mathbb R)}\Vert D_xu\Vert_{L^2(\mathbb R)} $ and thus $\Vert u\Vert_{L^2(\mathbb R)}\le \Vert D_xu\Vert_{L^2(\mathbb R)}$ proving that for functions whose support has a diameter $\le 1$ the $H^1$ norm is equivalent to the $L^2$ norm of the derivative. You get an answer for $s=1$, which can probably be extended to any $s>1/2$ (with different constants).

Answer 2

For $u \in H^2(I)\cap H^1_0 (I)$, $I=(1,2)$, the inequality is equivalent to $$\|(-\Delta)^{-s/2}v\|_\infty \le C \|v\|_2, \quad v \in L^2(I).$$ Let us use $$ (-\Delta)^{-s/2}v=\frac{1}{\Gamma (s/2)}\int_0^\infty t^{s/2 -1}T(t)v\, dt$$ where $T(t)$ is the semigroup generated by the Dirichlet Laplacian in $I$. If $\lambda_1$ is its first eigenvalue then $\|T(t)\|_{2 \to 2} \le e^{-\lambda_1 t}$ and $\|T(t)\|_{ 2 \to \infty} \le C_1t^{-1/4}$, by Gaussian estimates. The semigroup law yields $\|T(t)\|_{2 \to \infty} \le C_2 t^{-1/4}e^{-\lambda_1 t/2}$ and finally $$\|(-\Delta)^{-s/2} v\|_\infty \le \frac{C_2}{\Gamma (s/2)}\|v\|_2 \int_0^\infty t^{s/2-5/4}e^{-\lambda_1 t/2}\, dt \le C_3 \|v\|_2$$ if $1/2<s<2$. Finally, note that all constants $C_i$ can be computed since $\lambda_1$ is known and the Gaussian estimates follow from domination with the heat semigroup in the whole space.